Weight painting stuck in some kind of weird mode
This is a mode called Paint Mask. While in Weight Paint mode, go up and look at the top left of your 3D View, and you'll see a button with a face icon on it. You will likely find that it is activated. Turn it off to go back to normal.

The way that this mode works is that whatever faces you select while in Edit Mode will be the only faces that can receive a brush effect while in Weight Paint mode. It's very handy in this regard. A good feature, to be sure. But it can be activated by accident, and lead to frustration. If you go into Edit Mode for the object you're trying to weight paint, you will probably see that the same face that is showing up blue in the pic you provided is already selected. Even if you unselect these faces, though, you'll still need to turn off the paint mask in Weight Paint mode. If no faces are selected in Edit Mode, and the paint mask is still on in Weight Paint mode, then you won't  see any weird blue faces, but you also won't be able to paint at all.